Summary: The vector partition problem for convex objective
functions
Shmuel Onn # Leonard J. Schulman +
Abstract
The partition problem concerns the partitioning of a given set of n vectors in d­space into
p parts so as to maximize an objective function which is convex on the sum of vectors in each
part. The problem has broad expressive power and captures NP­hard problems even if either
p or d is fixed. In this article we show that when both p, d are fixed, the problem is solvable in
strongly polynomial time using O(n d(p-1)-1 ) arithmetic operations. This improves upon the
previously known bound of O(n dp 2
). Our method is based on the introduction of the signing
zonotope of a set of points in space. We study this object, which is of interest in its own right,
and show that it is a refinement of the so called partition polytope of the same set of points.
Mathematics Subject Classifications: 05A, 15A, 51M, 52A, 52B, 52C, 68Q, 68R, 68U,
90B, 90C
Keywords: partition, cluster, optimization, separation, convex, polytope, zonotope, sign­
ing, vertex enumeration, polynomial time, combinatorial optimization
1 Introduction
The partition problem concerns the partitioning of a set A = {a 1 , . . . , a n } of n vectors in d­space
into p parts so as to maximize an objective function which is convex on the sum of vectors in